LEAPS-MPS Where Algebra and Topology Meet: An investigation in Braid Group Representation Theory

Knots play an important role in the world around us in a much deeper way than just your electronic chords tangling under your desk. DNA molecules are knotted; the paths of satellites form interwoven orbits around earth; quantum computing can be modeled by the braiding of particles. This project focuses on different ways of storing the crossing information and tangled structure of knots in a format that can be computationally processed. This is called a representation- choosing different mathematical ways to represent, or encode, 3 dimensional knotted information. As both a mathematician and dancer, the PI uses dance to help explain her research visually and recruit the next generation of math pioneers. This project focuses on representation of a specific type of knotted structure called a braid. Topologically, a braid is a tangle of n-strands that flow monotonically from the floor to the ceiling. The set of all n-stranded braids can be turned into a group using vertical stacking as the multiplication operation, rendering an algebraic description of braids. To describe a meaningful representation requires constant negotiation between the topological and algebraic descriptions of braids. The research agenda of this project centers on constructing and analyzing representations of braids in three separate directions. Firstly, the PI will find a topological description (in terms of mapping class groups) of an algebraic generalization of the braid group, called the virtual braid group. This will be done by extending a construction on a finite index subgroup of the virtual braid group to the entire group via spanning surfaces. Secondly, using a mapping class group representation of the braid group, the PI will classify when certain link diagrams have a bi-orderable complement. Thirdly, the PI will alter a step in a well known construction for building braid group representations (the Long-Moody construction) and thereby create a much larger 1-parameter family of representations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.